Vectors & 3D Geometry PYQ — Page 13

Let $ABC$ be a triangle such that $\vec{BC}=\vec{a},\vec{CA}=\vec{b}$, $\vec{AB}=\vec{c},|\vec{a}|=6\sqrt{2},|\vec{b}|=2\sqrt{3}$ and $\vec{b}\cdot \vec{c}=12$ Consider the statements : $(S1)$: $|(\vec{a}\times \vec{b})+(\vec{c}\times \vec{b})|-|\vec{c}|=6(2\sqrt{2}-1)$ $(S2)$: $\angle ABC={\mathrm{cos}}^{-1}(\sqrt{\frac{2}{3}})$. Then

Let $\vec{a}=2\hat{i}-\hat{j}+5\hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2\hat{k}$. If $((\vec{a}\times \vec{b})\times \hat{i})\cdot \hat{k}=\frac{23}{2}$, then $|\vec{b}\times 2\hat{j}|$ is equal to

Let $A,B,C$ be three points whose position vectors respectively are: $\vec{a}=\hat{i}+4\hat{j}+3\hat{k}$ $\vec{b}=2\hat{i}+\alpha \hat{j}+4\hat{k},\alpha \in R$ $\vec{c}=3\hat{i}-2\hat{j}+5\hat{k}$ If $\alpha$ is the smallest positive integer for which $\vec{a},\vec{b},\vec{c}$ are non-collinear, then the length of the median, $\triangle ABC$, through $A$ is:

If $\vec{a}=2\hat{i}+\hat{j}+3\hat{k},\vec{b}=3\hat{i}+3\hat{j}+\hat{k}$ and $\vec{c}={c}_{1}\hat{i}+{c}_{2}\hat{j}+{c}_{3}\hat{k}$ are coplanar vectors and $\vec{a}\cdot \vec{c}=5,\vec{b}\perp \vec{c}$, then $122({c}_{1}+{c}_{2}+{c}_{3})$ is equal to ______.

Let $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+2\hat{k}$. Then the number of vectors $\vec{b}$ such that $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{b}|\in {1,2,\ldots ,10}$ is

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times (\vec{b}+\vec{c})=\vec{0}$, then the value of $3(\vec{c}.\vec{a})$ is equal to _______.

Let $\vec{a}=\alpha \hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=-2\hat{i}+\alpha \hat{j}+\hat{k}$, where $\alpha \in R$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15({\alpha }^{2}+4)}$, then the value of $2{|\vec{a}|}^{2}+(\vec{a}\cdot \vec{b}){|\vec{b}|}^{2}$ is equal to

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the points $A$ and $B$. Then ${(AB)}^{2}$ is equal to

Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3\hat{i}-5\hat{j}+4\hat{k}$ be two vectors, such that $\vec{a}\times \vec{b}=-\hat{i}+9\hat{i}+12\hat{k}$. Then the projection of $\vec{b}-2\vec{a}$ on $\vec{b}+\vec{a}$ is equal to

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$, If $|\vec{a}\times \vec{b}|=|\vec{a}|$, then the angle between the vectors $(\vec{a}+\vec{b}+(\vec{a}\times \vec{b}))$ and $\vec{a}$ is equal to :

The angle between vectors a=i+j and b=j+k is:

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $(\vec{a}\times \vec{b})$ and $\vec{c}$ is $\frac{\pi }{6}$, then the value of $|(\vec{a}\times \vec{b})\times \vec{c}|$ is:

Let the vectors $(2+a+b)\hat{i}+(a+2b+c)\hat{j}-(b+c)\hat{k},(1+b)\hat{i}+2b\hat{j}-b\hat{k}$ and $(2+b)\hat{i}+2b\hat{j}+(1-b)\hat{k},\forall a,b,c\in R$ be co-planar. Then which of the following is true?

In a triangle $ABC,$ if $|\vec{BC}|=3,|\vec{CA}|=5$ and $|\vec{BA}|=7,$ then the projection of the vector $\vec{BA}$ on $\vec{BC}$ is equal to

Let $\vec{c}$ be a vector perpendicular to the vectors $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. If $\vec{c}\cdot (\hat{i}+\hat{j}+3\hat{k})=8$, then the value of $\vec{c}\cdot (\vec{a}\times \vec{b})$ is equal to

The lines $x=ay-1=z-2$ and $x=3y-2=bz-2,(ab\neq 0)$ are coplanar, if:

The angle between the straight lines, whose direction cosines $l,m,n$ are given by the equations $2l+2m-n=0$ and $mn+nl+lm=0$, is:

Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}.$ Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}.$ If $\vec{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\vec{a}$ is $19$ units, then $|2\vec{v}{|}^{2}$ is equal to _____.

A hall has a square floor of dimension $10m\times 10m$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is ${\mathrm{cos}}^{-1}\frac{1}{5},$ then the height of the hall (in meters) is: 

Let $\vec{a}=\hat{i}+5\hat{j}+\alpha \hat{k},\vec{b}=\hat{i}+3\hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that, $|\vec{b}\times \vec{c}|=5\sqrt{3}$ and $\vec{a}$ is perpendicular to $\vec{b}.$ Then the greatest amongst the values of $|\vec{a}{|}^{2}$ is ________.

Let $a,b$ and $c$ be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are co-planar, then $c$ is equal to:

Let $\vec{a}=\hat{i}+2\hat{j}-\hat{k},\vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=\vec{c}\times \vec{a}$ and $\vec{r}\cdot \vec{b}=0,$ then $\vec{r}\cdot \vec{a}$ is equal to

Let $\vec{a},\vec{b},\vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36{\mathrm{cos}}^{2}2\theta$ is equal to

In a triangle $ABC$, if $|\vec{BC}|=8,|\vec{CA}|=7,|\vec{AB}|=10$, then the projection of the vector $\vec{AB}$ on $\vec{AC}$ is equal to :